Fundamental Theorem of Calculus I
Differentiation undoes integration.
Key idea
Differentiation undoes integration.
In depth
If F(x) = ∫ₐˣ f(t) dt, then F′(x) = f(x). Differentiation undoes integration: the rate at which the accumulated area grows equals the height of the curve.
Worked example
d/dx ∫₁ˣ t² dt = x². With a chain: d/dx ∫₀^(x²) sin t dt = 2x sin(x²).
Formula
Real-world uses
- Linking the rate of water flowing into a tank to the total water stored
Common mistake
Forgetting the chain rule when the upper limit is a function of x.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Fundamental Theorem of Calculus I?
Differentiation undoes integration. It is a topic at the expert level in Calculus.
What is the formula for Fundamental Theorem of Calculus I?
The key formula is d/dx ∫ₐˣ f(t)dt = f(x).
Where is Fundamental Theorem of Calculus I used in real life?
Real-world examples: Linking the rate of water flowing into a tank to the total water stored.
What should I learn before Fundamental Theorem of Calculus I?
Make sure you are comfortable with Definite Integral, the previous topic in Calculus.
What is a common mistake with Fundamental Theorem of Calculus I?
Forgetting the chain rule when the upper limit is a function of x.
How can I practise Fundamental Theorem of Calculus I?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Fundamental Theorem of Calculus I?
Move on to Fundamental Theorem of Calculus II next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.